Adjoint-based passive optimization of a micro T-mixer

POLITECNICO DI MILANO

Facolt`a di Ingegneria Industriale e dell’Informazione Corso di Laurea Magistrale in Ingegneria Aeronautica

Adjoint-Based Passive Optimization

of a Micro T-mixer

Dipartimento di Scienze e Tecnologie Aerospaziali Politecnico di Milano, Milano, Italy

Relatore: Prof. Maurizio Quadrio

Tesi di Laurea di: Roberto Mosca, matricola 838984

Sommario

In questo lavoro viene presentato un processo di ottimizzazione, basato sul metodo aggiunto, delle caratteristiche di mescolamento di un micro T-mixer. Oltre all’interesse tecnologico del problema specifico, la procedura intende anche valutare l’usabilit`a pratica e la robustezza delle tecniche aggiunte nel contesto di librerie CFD a volumi finiti esistenti per problemi di flussi incomprimibili. In questo senso, la scelta di un caso geometricamente semplice come quello del T-mixer `e da considerarsi naturale, essendo il presente lavoro un primo passo verso l’applicazione a casi pi`u complessi. Un’ulteriore semplificazione del problema `e data dalla natura lam-inare del regime di moto considerato. L’approccio utilizzato `e quello dell’ottimizzazione di forma, da considerarsi come un caso parti-colare dell’ottimizzazione topologica. Una formulazione continua viene adottata; a seguito dell’introduzione di una specifica fun-zione costo per il problema in esame, le equazioni aggiunte e le relative condizioni al contorno sono state implementate mediante l’utilizzo del software open-source OpenFOAM allo scopo di cal-colare la sensitivit`a superficiale. L’ottimizzazione `e applicata in uno specifico regime laminare, denominato vortex flow regime, dove il dispositivo presenta un’insufficiente capacit`a di mescolamento. In questo contesto gli effetti della diffusione numerica portano a sovrastimare il grado di mescolamento. Per questo motivo diverse

simulazioni numeriche su griglie differenti sono state effettuate per tenere sotto controllo gli effetti della diffusione numerica; una soglia, corrispondente ad una dimensione della cella di 3.0 µm, `e stata identificata come la dimensione massima al di sotto della quale la soluzione non presenta cambiamenti qualitativi e i risultati dipen-dono marginalmente dalla risoluzione spaziale. Attraverso un solo passo dell’ottimizzazione e cambiamenti minimi della geometria, `e stato possibile incrementare il grado di mescolamento di almeno due ordini di grandezza, mentre la caduta di pressione `e rimasta pressoch´e invariata.

Abstract

Adjoint-based optimization is used to passively improve the mixing characteristics of a micro T-mixer thanks to the design of an im-proved geometry of the device. Besides the technological interest of the specific problem, the procedure is also meant to assess the practical usability and robustness of adjoint techniques in the con-text of existing finite-volumes CFD libraries for incompressible flow problems. The choice of a geometrically simple case, like that of the T-mixer, should be considered natural, being this work a first step towards more complex applications. A futher simplification is given by the laminar nature of the flow regime taken into account. A continuous adjoint formulation is adopted; after the introduction of a properly selected objective function for the problem at hand, adjoint equations and boundary conditions are implemented in the open-source software OpenFOAM to compute the surface sensitiv-ity to the degree of mixing. We specifically target the laminar flow regime (i.e. the vortex flow regime) where the T-mixer shows a low mixing efficiency. It is well know that, in this regime, the numerical diffusion affects the results, and in particular leads to an overesti-mate of the degree of mixing. Several calculations on meshes of different size are used to keep resolution effects under control; a threshold, corresponding to cell lenght of 3.0 µm, is identified, be-low which the solution does not present further qualitative changes,

and the results are marginally affected by further increases in spa-tial resolution. With only one optimization step, and rather small geometry changes, we are able to increase the degre of mixing by more than two orders of magnitude, while the pressure drop across the device is basically unchanged.

Ringraziamenti

Il ringraziamento pi`u grande lo dedico alla mia famiglia. A Francesca e Gianni che mi hanno supportato in tutti questi anni senza mai farmi mancare il loro sostegno anche nei momenti pi`u difficili. A Rebecca, per il suo sincero e costante sostegno e interesse per la mia vita.

Un profondo ringraziamento a Davide, compagno di universit`a ma prima ancora di vita, a Simone e Yuri, a cui sono legato da un pronfondo affetto, a Davide, amico lontano, e infine ma non ultimo ad Andrea, per le chiacchierate fino a tarda notte.

Grazie al Prof. Quadrio che ha sempre saputo spronarmi e moti-varmi nel portare avanti questo lavoro, `e stato un piacere e un onore poter lavorare con lui.

Un ringraziamento a tutte le persone che sono state presenti nella mia vita nei momenti felici ma anche in quelli pi`u difficili.

Contents

1.3 Structure of the Thesis . . . 13

2 Problem Set Up 15

2.1 Geometry And Governing Equations . . . 15

2.2 Mixing . . . 17

2.3 The adjoint equations . . . 19

3 Workflow 25

3.1 Optimization Procedure . . . 25

3.2 Solvers and Numerical Schemes . . . 26

3.3 The cost function . . . 29

4 Results 33

4.2 Optimized T-mixer . . . 38 4.3 Random Bump . . . 44 5 Conclusions 47 Bibliography 49 A Tables 57 2

List of Figures

1.1 Sketch of a typical T-shape micromixer with the ref-erence coordinate system. . . 8

4.1 Streamlines view from the back wall (positive x-axis) 34

4.2 Streamlines view from superior surface (negative z-axis) . . . 34

4.3 Degree of mixing trend with respect to the number of cells . . . 35

4.4 Concentration field at the outlet of the mixing chan-nel (x = 3000 µm) . . . 36

4.5 Surface sensitivity map at the back wall (15M cells mesh) . . . 38

4.6 Optimized geometry: view of the back wall for a 8 µm bump (15M cells mesh) . . . 38

4.7 Concentration field at the outlet of the modified mix-ing channel (x = 3000 µm) for 4, 8, 12, 16 and 20 µm bump size (15M cells mesh) . . . 39

4.8 Streamlines view from the back wall of the optimized geometry (positive x-axis) . . . 40

4.9 Degree of mixing as function of the bump size for the 15M cells mesh . . . 40

4.10 Degree of mixing of the optimized T-mixer as func-tion of the number of cells for a 12 µm bump size . . 41

4.11 Degree of mixing trend along the mixing channel for orginal, 4, 8, 12, 16 and 20 bump size geometries . . 42

4.12 Degree of mixing trend at the entrance of the mixing channel for orginal, 4, 8, 12, 16 and 20 bump size geometries . . . 43

4.13 View of the back wall for the random deformation for a 4µm bump size . . . 44

4.14 Degree of mixing trend along the mixing channel for the orginal and random geometries . . . 44

List of Tables

A.1 Mesh Resolution Results . . . 57

Chapter 1

Introduction

Well she’s walking through the clouds With a circus mind that’s running round Butterflies and zebras

And moonbeams and fairy tales That’s all she ever thinks about Riding with the wind

Little Wing

Microfluidics devices have seen in recent years a considerable in-crease of interest in many fields, including analytical chemistry and life sciences. Among such devices, micromixers are important owing to their potential for effective and rapid mixing of samples. Applica-tions are rapidly increasing, and presently range from rapid mixing of macromolecular solutions for chip-based molecular diagnostics to the fast-growing market of biochips, with application spanning DNA sequencing, sample preparation and analysis, cell separation and detection, as well as environmental monitoring. The potential of miniaturized mixing devices is clearly related to the small re-quired amount of samples and reagents, their rapidity in mixing,

8 Chapter 1. Introduction

Figure 1.1: Sketch of a typical T-shape micromixer with the reference coordinate system.

their low unit cost and high throughput. The interest in such de-vices is high, for both academic and industrial fields, and several review papers are available, among which we refer the interested reader to [1–3] and the most recent [4]. In a fluid system the ra-tio of convective to diffusional mass transport is quantified by the P´eclet number P e = U h/D, where U is a typical flow velocity, h is a representative cross-stream length-scale and D is the diffusion coefficient. In the typical microchannel that is present in micromix-ers, P e ranges from 101 to 105 [5], indicating that in such system convection is faster than molecular diffusion [6]. Thus, in spite of the small geometrical dimensions of the system, molecular diffusion does not significantly contribute to smoothing out concentration gradients down to the smallest scales in a reasonable amount of time. Fast and complete mixing in a microchannel can not be

ob-9

tained trivially, in fact the Reynolds number (Re) that characterizes the flow in such devices often assumes small values, therefore tur-bulent mixing can not take place. As the effectiveness of mixing is a key technological factor to enable development of smaller de-vices, optimization and performance tuning of micromixers is an important technological task.

One of the most extensively studied devices is the so-called T-mixer (Fig.1.1), because of its geometrical simplicity [7]. It is com-posed by two straight ducts, connected by a T-junction, carrying two different streams which gradually mix together while flowing along the outlet channel. In the basic configuration, the ducts have a constant square or rectangular cross-section whose dimensions, the width (W ) and height (H), vary in the literature between 50 and 500 µm, respectively [6,8]. The flow in a T-mixer is known to possess three laminar regimes [6,7,9]. As described in [10], the onset of the various regimes depends on the parameters of the sys-tem, that can be assembled into a so-called identification number K, defined as K = Re0.82 W 0 H −0.79  Dh,in D_h0 −1.5  W H 0.15 , (1.1)

where Dh,in and D0h are the hydraulic diameters at the inlet and

outlet ducts, respectively, with the general hydraulic diameter de-fined as Dh = 4A/P , where A is the cross sectional area and P is

the wetted perimeter of the section, and W0 identifies the outlet width. Once the geometry is fixed, the main parameter becomes Re. At very low Re the two incoming streams remain segregated, convection is ineffective and only molecolar diffusion is at work to promote mixing [11]. The streamlines after the junction are almost straight: this is the so-called segregated regime, for which mixing efficiency is very low. At intermediate values of Re, in the so-called vortex flow regime, two pairs of vortices develop at the entrance of

10 Chapter 1. Introduction

the mixing channel, but they maintain axial symmetry w.r.t. the duct centerline. Mixing improves thanks to the vortical advection of the two fluids; however higher Re imply high velocities, hence a smaller residence time for the fluid particles to diffuse [12]. Over-all, mixing efficiency in the vortex flow regime is only mildly im-proved. At even higher Re, in the so-called engulfment regime, the axial symmetry is broken, and the quality of the achieved mixing strongly increases. At odds with the previously mentioned regimes, which are steady, unsteady regimes can also take place as recently reported in [9,13]. The numerical values of Re marking the passage from one regime to another are dependent upon the geometry of the specific T-mixer [14].

1.1

State of Art

It is of utmost interest to achieve high mixing within a short dis-tance. This makes the engulfment regime the most attractive one for applications, however it is desiderable to obtain the highest mix-ing quality with the lowest flow speed, hence the target is to achieve good mixing in the segregated or vortex-dominated regime. Sev-eral papers exist which describe attempts to improve the mixing efficiency in micromixers. The various strategies can be classified into passive and active: the former refers to the modification of the basic geometrical configuration of the device, while the lat-ter adds exlat-ternal forcing to increase mixing [1–3]. Since mixing relies mainly on molecular diffusion and chaotic advection, all the optimization attemps focus on increasing these mass transport phe-nomena. Molecular diffusion is positively affected by increasing the contact surface area between the working fluids and decreasing the diffusion path. This, in the case of micromixers based on serial or parallel lamination process, is obtained by splitting the streams into multiple substreams and recombining them along the mixing

1.1. State of Art 11

channel. The most basic T-mixer and Y-mixer belong to this cat-egory. A different type of parallel lamination is obtained in the injection mixer, formed by an array of nozzles injecting solvent in the solute flow. Because of the low Re numbers involved, the flow is laminar, therefore the turbulent mixing does not take place and the streams mix mainly due to diffusion transport in a very slow process. Advection occurs only in the main flow direction and it does not contribute to the transversal mass transport helping the streams mixing [2]. In the literature many attemps to generate chaotic advection have been described, many with the adoption of obstacles of different sizes and shapes such as grooves [15,16], rect-angular, rhombic, semicircular and circular 3D objects [17–20], or designing split and recombine (SAR) geometries, zig-zag or curved micromixers [21,22] and convergent-divergent channels [23], which all enhance the contact surface area or produce recirculations at the corners. Also the generation of droplets by capillary effect, causing secondary internal flows, is exploited to enhance chaotic advection. On contrary, active micromixers introduce a disturbance through energy inputs of different nature. Simplest active micromixers dis-turbs the pressure field not by injecting the liquid continuosly but introducing a pulsating flow rate at the inlets [24,25]. Acustic per-turbations have been experienced in order to generate disturbances in the pressure field or bubbles via cavitation [26,27]. Electro-hydrodynamic instability (EHD) and electro-osmotic flow (EOF) are exploited through electrical field produced by electrodes caus-ing disturbances at the interface of the electrically charged fluids or vortices [28,29]. Finally, magnetohydrodynamics (MHD) effects are exploited in special ferrofluid phases [30,31]. Active micromix-ers are, generally, characterized by more complex structures, thus requiring complex fabrication processes and power sources. On the other hand passive micromixers are simple, robust and stable in operation and don’t need energy external sources except for that

12 Chapter 1. Introduction

for fluid delivery.

1.2

Adjoint Method

Being our application a first step to enhance mixing in a micromixer through the adjoint method, the choice of a simple geometry, such that characterizing the passive T-mixer, should be considered nat-ural. All the optimization attemps aforementioned are based on a trial and error approach and a qualitative analysis of the phe-nomena involved. In this work, we have described the potential of a passive strategy to optimize the T-mixer geometry that goes beyond the empirical approaches described above, and aims at ra-tionally determining the optimal shape of the back wall of the two inlet ducts (y−z plane at x = 0 with reference to Fig.1.1) that maxi-mizes mixing along the outflow duct. Shape optimization problems are often encountered in Computational Fluid Dynamics and in particular in aerospace applications [32]. Gradient-based optimiza-tion method focus on the so-called numerical sensitivity analysis, which consists in determining the gradients of a defined cost func-tion w.r.t. a set of design variables, subsequently used to reduce the cost function itself. The techniques available for calculating the sensitivity derivatives are generally classified into direct and adjoint techniques [33]. The direct approach includes method such as finite differencing, direct differentiation and complex variable method [34]. On the other hand, in the past decades, the adjoint approach has recieved the most of the attentions, as the cost of calculating the gradients is almost indipendent from the number of design variables [35]. This is highly advantageous when the number of design variables is large [36], as in the present case, where they represent a parametrized description of the shape of the back wall. The adjoint method introduces auxiliary, or adjoint, variables cal-culated through the solution of a linearized form of the governing

1.3. Structure of the Thesis 13

equations, in this case the Navier–Stokes equations, and allows to calculate the surface sensitivity through the solution of only one ex-tra linear system [34]. To our knowledge, the only papers so far that used adjoint Navier–Stokes equations in the context of a T-mixer are those by Fani, Camarri and Salvetti [9,13]. In the present case, to increase simplicity and aim at an entirely passive device, we will only consider as control variable the geometry of the device, so that it will produce an optimized geometry by the use of the so-called surface sensitivity.

1.3

Structure of the Thesis

The structure of the thesis is as follows. In chapter2, the geometry, dimensions and set up of the problem are presented together with the governing equations for both the direct and adjoint problems. In chapter 3 the derivation of the adjoint boundary conditions, their implementation and definition of the cost function are reported. Results and conclusions are discussed in chapters 4and 5.

Chapter 2

Problem Set Up

2.1

Geometry And Governing Equations

The geometry considered in this work is that represented in Fig.1.1

and the set up replicates the one used by Galletti et al.[37]. In particular, the inlet ducts have constant rectangular cross-section with width and height equal to W = 150 µm and H = 200 µm, respectively, and length Lin= 1500 µm. Dimensions of the mixing

channel are W0 = 300 µm and Lout = 3000 µm. The fluid motion

is governed by the three-dimensional incompressible Navier–Stokes equations which, in dimensionless form, read:

∇ · v = 0 (2.1) ∂v ∂t + (v · ∇) v + ∇p = 1 Re∇ 2_{v , (2.2)} where Re is defined as Re = DhU ν (2.3)

and the mean inlet velocity is U = 0.42 m/s, yelding Re = 100 defined on the hydraulic diameter of the outlet channel. This value

16 Chapter 2. Problem Set Up

corresponds to the steady vortex regime, where the mixing is poor due to the limited diffusion combined with short residence time, and an increase in mixing performance is required. This scenario allows a better understanding of the capacities of the adjoint method. No-slip and zero-gradient boundary conditions are imposed at the wall and outlet respectively, instead an in-depth analysis is necessary for the inlet boundary conditions. Galletti et al.[37] demonstrated a dependency of the solution in the two cases the flow enters in a fully developed condition (i.e. FD case) or if the flow is injected with uniform normal velocity (i.e. non-FD case). Results differences are significant in the engulment regime, because a FD condition requires longer developement region than the inlet duct; on contrary they are negligible in the stratified and vortex regime, since the flow completely develops before reaching the junction. Eq.2.4shows the relation proposed by Dombrowsky et al.[38] to estimate the duct length, for rectangular sections, nedeed for the flow to reach fully developed condition.

Le

Dhin

= 0.379exp (−0.148Rein) + 0.055Rein+ 0.26 (2.4)

In the case of Re = 100, it predicts Le/Dhin∼= 4.20, thus Le= 700

µm which is less than the inlet duct length. Scholars argued that, for inlet velocities such that U ≤ 0.8 m/s, the velocity profiles at the entrance of the mixing channel are similar for both FD and non-FD cases [37]. However we considered a fully developed boundary condition at the inlets for our case. To limit the computational burden and, in the same time, to achieve a proper inlet condition, one can resort, as done in [39,40], to a series expansion for a quasi-analytical expression of the inflow. In alternative, as done here, an auxiliary transient simulation can be performed in a straight chan-nel, long Le = 300 µm, by imposing periodic boundary conditions

2.2. Mixing 17

flow rate. Once the flow becomes fully developed, i.e. indipendent from the longitudinal direction, the outlet velocity field is taken as boundary condition at the T-mixer inlets. The velocity boundary conditions are summarized in the following

v = v Γin (2.5)

v = 0 Γw (2.6)

(∇v) · n = 0 Γout (2.7)

where Γin, Γw and Γout refer to the inlets, wall and outlet patches

respectively, and v to the fully developed velocity distribution. Fi-nally, the pressure field satisfies a zero-gradient boundary condition at the inlets and at the wall, with a null value assigned at the outlet:

(∇p) · n = 0 Γin, Γw (2.8)

p = 0 Γout. (2.9)

Now the boundary conditions for the primal problem are complete.

2.2

Mixing

To model the mixing process of the two streams that flow through the T-mixer, we consider a passive scalar, possessing uniform con-centration c = 0 at one inlet section and c = 1 at the other. The following evolution equation for the passive scalar describes how concentration changes in the computational domain:

∂c

∂t+ ∇ · (vc) = 1 ReSc∇

2_{c , (2.10)}

where Sc = ν/D represents the Schimdt number with ν the kine-matic viscosity of the fluid and D the molecular diffusivity. We consider water at ambient temperature (density ρ = 998 kgm−3, dynamic viscosity µ = 10−3 kgm−1s−1 and kinematic viscosity

18 Chapter 2. Problem Set Up

ν = 1.02 · 10−6m2s−1) as working fluid. The molecular diffusivity of the passive scalar, corresponding to violet dye, is D = 3 × 10−10 m2/s, so that diffusion is negligible compared to convection (i.e. large P e) in determining mass transport. As discussed by Ottino and Wiggins [5], this amounts to considering two different streams of liquid, identified by the concentration value c = 0 or c = 1. It should be mentioned that this approach implies the existence of very thin boundary layers separating the two liquid streams, and this calls for a correspondingly large resolution of the computational mesh. This supports (see chapter 4) our grid-convergence study.

An index describing the mixing efficiency needs to be defined. As for the geometry set-up adopted, we have retraced the steps followed by Galletti et al.[37]. Let be A the cross sectional area of the channel and vn the normal velocity component w.r.t. the

section. We first introduce a velocity-wieght mean concentration, the bulk concentration c, as

c = hcvni hv_ni hvni = 1 A Z A vndA hcvni = 1 A Z A cvndA , (2.11)

where the brackets indicates the non-weighted integral operator over a cross-section of area A. The quantity c represents the area-averaged mean value of the concentration weighted by the flow ve-locity. Because of its structure, regions with higher velocities, such as the center of the channel, contribute more than regions near wall. In our case, thanks to the symmetry of the two input chan-nels which possess equal flow rates, c = 0.5 in every normal section of the mixer. Second, a local weighted concentration variance and its normalized counterpart are defined

(∆c)2₌ (c − c) 2_v n  hvni σ_cm2 = (∆c) 2 (∆c)2 max , (2.12)

where the normalization factor is the maximum weighted concen-tration variance, which is obtained when the two streams are

com-2.3. The adjoint equations 19

pletely unmixed and corresponds to (∆c)2

max = 1/4. We finally

define the degree of mixing

δm= 1 − σcm. (2.13)

The degree of mixing goes from δm = 0, for entirely unmixed

streams, to δm = 1, for perfectly mixed streams. At the outlet,

in the vortex regime δm = O 10−2
, while in the engulment regime

δm= O 10−1
as reported in [37].

2.3

The adjoint equations

In the following, the theory behind the surface-sensitivity approach has been very briefly recalled; several documents are available to provide a general introduction to adjoint techniques as a tool for solving optimization problems [41], as well as more specific refer-ences [42]. What follows is closely derived from [43,44], which pre-sented a detailed derivation of both topological and surface sensitiv-ity for the case of internal flows. In this work, a continuous-adjoint approach is followed. This implies that, starting from the linearized primal equations in their analytical form, the adjoint equations and boundary conditions are derived analitycally and then discretized, as it is by far the easiest approach when the adjoint solver must be based on an existing finite-volume library of tools like OpenFoam [44]. Computing surface sensitivities can be interpreted as a variant of the more general topological optimization approach where only wall-normal deformations are considered as control variables [43].

Let J be an arbitrary cost function, depending on velocity and pressure fields, that will be later specified based on phyisical insight of the particular problem at hand and the related specific techno-logical optimization goal. A constrained optimization problem is written as a minimization problem for J under the constraint that

20 Chapter 2. Problem Set Up

velocity v and pressure p satisfy the Navier–Stokes equations, i.e. minimize J = J (v, p; β) subject to R (v, p) = 0 , (2.14) where R = (R1, R2, R3, R4)T stands for the governing equations

(Eq.2.1and2.2) and β is an infinitesimal localized normal displace-ment of the domain boundary. The problem is now reformulated by defining a Lagrange functional L as

L , J + Z

Ω

(u, q) RdΩ , (2.15)

where u = (u1, u2, u3)T and q are the lagrange multipliers used to

enforce the constraints and will be referred to in the following as the adjoint velocity and adjoint pressure fields. In order to evaluate the sensitivity of the cost function w.r.t. the design variable, the total variation of L, δL, is required.

δL = δβL + δvL + δpL . (2.16)

In this expression, beyond the explicit dependence of L on β, the two terms δvL and δpL represent the change of L due to the

vari-ation in the v and p fields as a consequence of the deformvari-ation of the domain. The Lagrange multipliers (u, q) can be chosen in such a way these terms vanish identically, i.e.

δvL + δpL = 0 . (2.17)

If this is the case, the variation δβL can be derived explicitly:

δL = δβL = δβJ +

Z

Ω

(u, q) δβR dΩ , (2.18)

which involves only the derivative w.r.t. β. Eq.2.17 represents the key semplification of the so-called adjoint method and allows to derive the adjoint equations for the adjoint variables u and q. Let

2.3. The adjoint equations 21

us now consider infinitesimal perturbations δv and δp around the reference values v and p, such that

v = v + δv (2.19)

p = p + δp . (2.20)

We now derive the needed variations, the linearized Navier–Stokes equations, and by neglecting the second-order terms, we obtain:

δv(R1, R2, R3)T = (δv · ∇) v + (v · ∇) δv − ν∇2δv (2.21)

δvR4 = −∇ · δv (2.22)

δp(R1, R2, R3)T = ∇δp (2.23)

δpR4= 0. (2.24)

By now decomposing J into a contribution JΩ, from a volume

in-tegral over the volume Ω, and a contribution JΓ, from a surface

integral over the boundary Γ of the volume, and by carrying out a procedure of integration by parts, we obtain the general form of the adjoint equations:

∇ · u = −∂JΩ

∂p (2.25)

− (u · ∇) v − (v · ∇) u = −∇q + ν∇2u − ∂JΩ

∂v . (2.26)

This is the general form of the adjoint equations system for the steady-state, incompressible Navier–Stokes equations system. It should be mentioned that variations of the viscosity ν are neglected. This is a correct assumption for laminar flow regime, as in our case, but, in the turbulent regime, it represents an approximation known as frozen turbulence [45]. The adjoint equations are linear w.r.t. the Navier–Stokes equations, the structure of the adjoint problem is very similar to that of the primal one. The most important difference is the minus sign in front of the convective term, that states the information is convected upstream of the primal flow

22 Chapter 2. Problem Set Up

rather than downstream [44]. Furthermore, in the rather common case where the objective function solely depends on the primal field at the boundary, JΩ = 0 because there is no contribution from

the internal domain fields, so the adjoint equations do not depend on J . Hence, the same numerical solver can be used for various optimization problems where the objective function changes. The adjoint equations state with Eq.2.25 that the field u is solenoidal; Eq.2.26 resembles the momentum equation, with p replaced by q, except that the convective term is now linear in the unknown u, and has a negative sign. A particular J will be reflected in the boundary conditions for the above equations. In the case of duct flows, where the boundary is typically composed by walls, inlet and outlet surfaces, the boundary conditions assume a relatively simple form provided that the fields δv and u are solenoidal with at least one of them being zero on Γ [44]. The adjoint variables at wall or at the inlet surface, where the primal velocity is typically assigned, must be specified as:

ut= 0 (2.27)

un= −

∂JΓ

∂p (2.28)

(∇p) · n = 0 (2.29)

where un and ut are the component of the adjoint velocity normal

and tangent to the boundary. The last condition stands from the observation that the adjoint pressure in Eq.(2.26) plays the same role of the pressure p in the primal momentum equation.

At the outlet surface, where the primal velocity typically obeys a zero-gradient condition, the boundary conditions assume the fol-lowing form 0 = vnut+ ν (n · ∇) ut+ ∂JΓ ∂vt (2.30) q = u · v + unvn+ ν (n · ∇) un+ ∂JΓ ∂vn , (2.31)

2.3. The adjoint equations 23

where the former specifies the parallel component of u, while the latter determines q. Let β be a field of wall-normal displacements of the wall boundary, defined in such a way that β > 0 is oriented as n and is positive from the fluid domain outwards. In Eq.2.18, the first term represents the explicit dependence of J upon β, which is zero when, as in the present case, β is defined on the walls while J contains expressions evaluated at inflow and/or outflow. Evaluat-ing the second term, where in this case the factor δβ can be writ-ten outside the integral, is more complex, as it apparently entails calculating changes induced by the surface deformations onto the primal solution. Luckily, the volume integral can be transformed into a boundary integral, and this term can be evaluated via a post-processing step once the primal and adjoint fields are known. This procedure, which is described for example in [46], makes use of the hypothesis JΩ= 0, because the cost function doesn’t contain

contri-bution from the volume, and that u = 0 at the wall. Moreover, the unknown variations δv and δp are approximated through a Taylor series expansion turcated to the first order as

δv = β (n · ∇) v + O(β2) (2.32)

δp = β (n · ∇) p + O(β2) . (2.33)

Finally the surface sensitivity to a normal displacement β over a little portion A of the wall surface can be written as

∂L

∂β = −Aν (n · ∇) ut· (n · ∇) vt, (2.34)

where the sensitivity information is contained into the product at the wall of the normal derivative of the tangential component of primal and adjoint velocity fields.

Chapter 3

Workflow

3.1

Optimization Procedure

To solve the governing equations, we used the open-source CFD package OpenFOAM, both to create the computational mesh and to post-process the simulations results [47]. The flow is laminar, the geometry and dimensions of the computational domain are skectched in Fig.1.1, and the primal case is designed to mimick the case at Re = 100 considered in [37]. Given the simple geometry, the computational mesh is created as a structured mesh through the utility blockMesh available within OpenFOAM. The grids are characterized by hexaedral elements of same dimensions and shape and, because the flow in a T-mixer is particularly sensitive to spa-tial resolution, we have carried out a careful study to assess the robustness of the results w.r.t. the cells number.

To solve the optimization problem, a suitable cost function first needs to be defined; boundary conditions for the adjoint problem are then set and compiled into the adjoint solver, that is run with the primal field as an input to calculate the adjoint fields. Now

26 Chapter 3. Workflow

surface sensitivity can be computed and used to build an improved version of the T-mixer geometry. By using the pointDisplacement tool, the new mesh can be constructed by deformation via assigning a field of point displacements. The performance of the optimized geometry is then tested with a further computation of the primal field over the modified geometry, and the subsequent passive scalar simulation.

The whole optimization process is thus made by the following steps:

1. Compute the primal solution.

2. Run the passive-scalar simulation to assess the performance of the T-mixer in terms of degree of mixing.

3. Compute the adjoint solution.

4. Compute the field of surface sensitivity, that requires primal and adjoint field.

5. Modify the geometry of the T-mixer exploiting the information of the sensitivity field.

6. Repeat points 1 and 2 above with the deformed geometry to assess performance improvements.

3.2

Solvers and Numerical Schemes

The solver used for the primal fields calculation is the unmodified simpleFoam, which solves the steady-state incompressible Navier-Stokes equations, both for laminar and turbulent regime, based on a Semi-Implicit Method for Pressure-Linked Equations (SIM-PLE ) algorithm [48]. SIMPLE is an iterative algorithm based on a segregated approach, which means that momentum and continuity

3.2. Solvers and Numerical Schemes 27

equations are not solved at the same time. At every iteration, after having set the boundary conditions and initialized the velocity and pressure fileds, a pesudo-momentum equation (predictor) is solved thanks to the gradient pressure term calculated using the pressure distribution from the previous iteration or an initial guess, to com-pute the intermediate velocity field, so-called because it doesn’t satisfy the continuity equation yet. Since the Navier-Stokes equa-tions don’t contain a proper equation for the pressure term, it’s necessary to take the divergence of the momentum equation and substitute it in the continuity equation. This passage gives us the equation, pressure corrector, needed to correct the pressure p. Now, the new velocity field can be calculated thanks to the new corrected pressure, and a new set of conservative fluxes satisfying the conti-nuity equation inferred. The procedure is finally repeated till a satisfactory level of convergence. Two under-relaxations limit the variables change from one iteration to the next; furthermore the pressure and momentum corrector can be repeated for a prescribed number of time to correct non-orthogonallity.

The adjoint solver is derived from the one created by Othmer [43,44] and present in the OpenFoam distribution. As the original one solves direct and adjoint equations iteratively for the topolog-ical sensitivity approach, we have created a more efficient version where the adjoint simulation is carried out only a posteriori, effec-tively decoupling it from the primal solver. The specified boundary conditions are adapted to reflect the particular problem considered here and derived below.

The goal is to perform a shape optimization and not a topo-logical one, therefore the adjoint equations have been modified by eliminating the Darcy’s term αv from the momentum equation. In Othmer’s solver, a criterion to evaluate if a cell has a positive or negative contribution, in terms of a specified cost function, lets to iteratively punish the counterproductive cells via a momentum loss

28 Chapter 3. Workflow

controlled by the Darcy’s term and proportional to the scalar α, named porosity. The domain is treated as a porous medium: where the porosity is high, the cell is penalized by an higher momentum loss threating it like a solid. The porosity is so an auxiliary vari-able which controls the transition from solid to liquid of the domain. Eliminating the momentum loss term, the adjoint governing equa-tions are the same of that of our problem. As for the direct problem, the adjoint system solution is based on a SIMPLE algorithm and, since the same differential terms are involved in both the primal and adjoint equations, the same second-order schemes have been utilized.

The passive scalar field is then calculated through the unmodi-fied solver scalarTransportFoam which solves Eq.2.10 after the ve-locity field calculation. In order to simulate the release of tracer at the inlets, an unsteady simulation has been performed over the steady velocity field precalculated. Since scalarTransportFoam does not guarantee boundedness of the scalar, the choice of a special bounded discretization scheme for the convective term is necessary to ensure the scalar between 0 and 1. The limitedLinear scheme, a second-order scheme that limits towards upwind in regions of rapidly changing gradient, is choosen to discretize the convective term and, following Jasak’s notation [49], the face flux φf flux can

be written as

φf = φU D+ ψ(φHO− φU D) , (3.1)

i.e. as the sum of the first-order bounded differencing scheme (UD) and a ”limited” high-order correction, where φHO is the face value

of φ for the selected high-order scheme and ψ is the flux limiter, in this case a Sweby limiter [50]. It requires a coefficient value between 0 and 1, with the highest corresponding to strongest limiting, and tending towards linear (i.e. central differencing) as the coefficient tends to 0. A first-order implicit scheme is assigned for the time

3.3. The cost function 29

derivative.

Soleymani et al.[51] suggested to apply a Quadratic Upstream Interpolation for Convective Kinematics (QUICK ) scheme, which is third-order, to discretize the convective term but, unfortunately, this produces unbounded solution [52]. To fix the problem, one could prefer the Upstream Monotonic Interpolation for Scalar Trans-port (UMIST ) scheme, which is a modified version of QUICK and provides bounded solution [53]. For the mesh morphing process, the utility moveDynamicMesh performs the mesh deformation by applying a normal displacement field at the nods of the back wall proportional to the cell sensitivity value, normalized in a [−1, +1] scale, and the maximum displacement admitted.

3.3

The cost function

As the passive-scalar approach decouples the calculation of the pri-mal velocity and pressure field from that of the concentration field, the obvious inclusion of the degree of mixing δm, or the

concentra-tion field c, into the objective funcconcentra-tion J is not feasible, since J may only depend on v, p and the design variable β. Hence, a deep understanding of the physical processes leading to mixing is a key step. This fact derives from the observation that what distinguish the mixing-efficient engulfment regime from the mixing-inefficient segregated or vortex flow regimes is the presence of significant large-scale rotation at the outlet. It is the rotational motion that moves fluid from one side of the duct to the other, thus reducing segre-gation. Hence we choose to work with an objective function that targets rotation in the outlet section:

J , Z Γout k kr × vtk2 dΓ , (3.2)

30 Chapter 3. Workflow

where r is the vector position with the origin in the center of the outlet section Γout, vt is the planar velocity on Γout and k = 1

m7/s5 is a unit constant to make the equation dimensionally cor-rect. The physical meaning of Eq.3.2 is clear, as its minimization entails maximizing the denominator, which is a measure of the solid-body rotation on Γout. Once J is defined, the derivatives needed

to explicitly write the boundary conditions of the adjoint problem, found in chapter 2, read as follows

∂JΓ ∂p = 0 (3.3) ∂JΓ ∂vn = 0 (3.4) ∂JΓ ∂vt = −2 n × r kr × vtk3 , (3.5)

and allow to write the boundary conditions for the adjoint equations with

u = 0 (3.6)

(∇p) · n = 0 (3.7)

at the walls and at the inlet patches. At the outlet the boundary conditions become

0 = vnut+ ν (n · ∇) ut− 2

n × r

kr × v_tk3 (3.8)

q = u · v + unvn+ ν (n · ∇) ut, (3.9)

and need to be implemented into OpenFOAM. Eq.3.9does not re-quire manipulations, since it contains explicitly q, so it can be im-plemented directly in its form

operator == ((Up & Uap) + nuw*snGradUan

3.3. The cost function 31

where Up & Uap stands for u · v, nuw*snGradUan is the normal gradient, and the last part corresponds to unvn, with the normal

velocities obtained from the adjoint and primal fluxes phiap and phip respectively. Boundary conditions both require the discretiza-tion of the gradient term, since it contains itself the variable u. Let be utw the variable value at the outlet patch and utw−1 the value

of the same variable at the adjiacent cells, the term (n · ∇) ut is

discretized with a first-order scheme: (n · ∇) ut'

utw − ut_w−1

h , (3.10)

where h is the distance between the centers of two neighboring cells in the direction perpendicular to the patch. Eq.3.8 can now be explicitly reformulated: utw = 1 vn+ ν/h  ν hutw−1 + 2 n × r kr × vtk3  . (3.11)

Not boundary condition is imposed on un by Eq.3.8, so it has no

constraints. The coded condition below corresponds to impose u = unn + ut and, through the first term of the equation, the normal

component is simply taken equal to itself. Eq.3.11 is implemented in the solver as follows:

vectorField::operator =

((phiap*patch().Sf())/(sqr(patch().magSf())) + (1/(Un + nd))*(nd*Uac_t + 2*(i^r)

/ ((pow((mag(r^Utangente)),3)) + SMALL)));

where the first term represents unn and SMALL identifies a special

Chapter 4

Results

In this chapter, simulations results are presented. All the simulaions have been carried out at the CINECA Computing Center on the system MARCONI.

4.1

Mesh Resolution

Fig.4.1shows a view of the streamlines from the back wall of the T-mixer. The flow field is, qualitatively, in agreement with the results present in the literature and is dominated by two pairs of counter-rotating vortices typical of the vortex regime. The increase in the mixing capacity w.r.t. the stratified regime, where the streamlines are straight, is due to the vortices at the entrance of the mixing channel, which produce a mass transport from the wall to the center of the mixing channel and viceversa. Nevertheless, as shown in Fig.4.2, the two streams still remain separated, flowing side by side along the outlet channel, and the mixing is only poorly improved.

Spatial resolution is known to affect simulations results, espe-cially in the context of liquid-liquid microdevices with

character-34 Chapter 4. Results

Figure 4.1: Streamlines view from the back wall (positive x-axis)

Figure 4.2: Streamlines view from superior surface (negative z-axis)

istic diffusion constant in the order of 10−9 m/s, where diffusive mass transport is dominated by numerical diffusion [54]. Since an in-depth preliminary characterization is needed to take spatial res-olution effects under control, we have carried out simulations using different structured grids, constitued, in ascending order, by 4, 7, 9.5, 12, 15, 20, 30 and 50 millions of cells, formed by cubic hexaedral elements of same shape and equal volume. A sudden jump charac-terizes the degree of mixing trend w.r.t the cells number between the 7M and 9M cells grids (Fig.4.3). This behaviour, justified by the visualization of the concentration field at the outlet (Fig.4.4),

4.1. Mesh Resolution 35 0.032 0.033 0.034 0.035 0.036 0.037 0.038 0.039 0.04 0.041 0.042 0.043 0 5 10 15 20 25 30 35 40 45 50 55 δm

Cells Number (in millions of cells)

Figure 4.3: Degree of mixing trend with respect to the number of cells

is attributable to the inadequate spatial resolution of the less re-fined grids. No mixing occurs in the right half plane for the 4M and 7M grids (Fig.4.4aand4.4b), while, starting from the 9M cells mesh (Fig.4.4c), the effects of the counter-rotating vortices, trans-porting mass from the center of the mixing channel to the wall, is visible also in the right half plane, leading to an increase of the degree of mixing. The jump can thus be related to the inability of the less refined grids to completely characterized the concentration field, which appears qualitatively similar for all the mesh forward (Fig.4.4d). A threshold of 3.0 µm is identified as the maximum cell length for grids able to correctly resolve the concentration field, confirming the estimate proposed by Roudgar et al.[6]. Below this threshold, corresponding in our case to the 9M cells mesh, δm has

phe-36 Chapter 4. Results

(a) 4M (b) 7M

(c) 9M (d) 50M

Figure 4.4: Concentration field at the outlet of the mixing channel (x = 3000 µm)

nomenon, the so-called numerical diffusion[51], which acts smearing out the sharpest concentration gradients and leading to artificial mixing and deviation of the numerical results [12]. Numerical dif-fusion plays a key role in this context and allows only a qualitative analysis of the flow field with the numerical results overstimating the mixing degree [55]. Different approaches to limit the numerical diffusion have been experienced in the context of a T-mixer. Soley-mani et al.[51], starting from a structured grid, refined the mesh in the area where the P´eclet number was too high. Bothe et al.[12] proposed an hybrid approach dividing the computational domain in a mixing and reaction zone, where a 3D and a 2D simulations are carried, respectively, by imposing a fully developed Poiseuille velocity profile in the mixing zone. This procedure, in the case of a reactive micromixer, leads to a grid independence for a minimum

4.1. Mesh Resolution 37

cell length of 0.3 µm. It is clearly impossible to apply this reso-lution to the entire computational domain, this justifies the use of the hybrid approach.

The degree of mixing and the pressure drop, defined as the pressure difference between the p value at the inlet and at the outlet sections, for the 4M cells mesh, the same used in [37], are δm =

0.0334 and ∆p = 1.46 kPa, respectively. They are in agreement with the results obtained by Galletti et al.[37] in the non-FD case, confirming that, in the vortex regime, a non-FD or FD condition at the inlets has a very little effect on the results. All the results for all the grids are reported in Tab.A.1, which also shows how the pressure drop is insensitive to the mesh resolution.

Results has let to define a minimum mesh resolution able to completely resolve the concentration field, but it has also pointed out how a quantitative solution, free from numerical diffusion, could be obtained only through the use of a huge amount of computational resources. In fact, despite the degree of mixing seems to converge to a stable solution, the difference between the last two points, with reference to Fig.4.3, still remains remarkable. Further grid refinements lead to a reduction of the numerical diffusion effects without any additional insight regarding the flow field.

In the following, the 15M cells mesh is so considered for all the subsequent simulations, as it provides the completely resolution of the flow field and allows to save computational resources w.r.t the more refined grids. In the present case, the 2.65µm length elements leads to 58 and 77 cells at the inlet channel for W and H respectively; the outlet channel is subdivided by 116 elements along the width W0 and by 1157 along the channel length Lout.

38 Chapter 4. Results

4.2

Optimized T-mixer

Figure 4.5: Surface sensitivity map at the back wall (15M cells mesh)

Figure 4.6: Optimized geometry: view of the back wall for a 8 µm bump (15M cells mesh)

After the primal field calculation, the adjoint problem has been per-formed to calculate the surface sensitivity at the back wall (Fig.4.5). The sensitivity map is symmetric due to the symmetry of the prob-lem and postive values of the sensitivity (red areas) indicate an increase of the cost function, so the mixing degree, for outwards displacements of the geometry. For negative values (blue areas) the effect of the deformation is clearly opposite. The mesh deformation process, referred as morphing, is carried out by applying a dis-placements field at the nodes proportional to the sensitivity value, normalized in a [−1, +1] scale, and a maximum displacement value. The bump size is a free design choice and, to indagate its effect, five different values, corresponding to 4, 8, 12, 16 and 20 µm, have been tested. Fig.4.6shows the T-mixer modified in accordance with the surface sensitivity.

In Fig.4.7the concetration field at the outlet of the modified T-mixer for the different bump sizes is presented. The passive scalar at the outlet assumes a typical S-shape distribution, more similar to that of the engulfment regime, with two main recirculations which become stronger with the increase of the bump size. The geometry deformation has as main effect the breaking of the symmetries of

4.2. Optimized T-mixer 39

(a) 4 µm bump (b) 8 µm bump

(c) 12 µm bump (d) 16 µm bump

(e) 20 µm bump

Figure 4.7: Concentration field at the outlet of the modified mixing channel (x = 3000 µm) for 4, 8, 12, 16 and 20 µm bump size (15M cells mesh)

the original flow field and, as Fig.4.8 shows, the two pairs of vor-tices at the junction make way for two main and bigger vorvor-tices responsible of the increased transverse mass transport, with the streamlines now crossing the mid plane of the outlet channel. The deformation produces a significantly improved degree of mixing: its trend, w.r.t. the bump size, is reported in Fig.4.9and shows an

40 Chapter 4. Results

Figure 4.8: Streamlines view from the back wall of the optimized geometry (positive x-axis) 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0 5 10 15 20 δm Bump Size [µm]

Figure 4.9: Degree of mixing as function of the bump size for the 15M cells mesh

high mixing capacity also for the lowest value of the bump size with δm = 0.0775, nearly twice the orginal degree of mixing. The trend

is almost linear, suggesting that a further significant increase of the mixing degree could be achieved for larger values of the bump size. Numerical results are reported in Tab.A.2.

4.2. Optimized T-mixer 41 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0 5 10 15 20 25 30 35 40 45 50 55 δm

Cells Number (in millions of cells)

Figure 4.10: Degree of mixing of the optimized T-mixer as function of the number of cells for a 12 µm bump size

carried out to the optimized geometry with an assigned bump of 12 µm. The results (Fig.4.10 and Tab.A.1) describes a monotonic descending trend, partially in contrast with the mesh independence study of the original geometry (Fig.4.3), indeed no jump occurs since the concentration field is completely resolved also for the less refined grids.

The optimization goal has been successfully achieved; the mix-ing capacity of the optimized T-mixer, not only is larger w.r.t. that of the original geometry in the vortex regime, but also w.r.t. the one in the engulment regime. Furthermore, the main advantage is reprensented by the little pressure drop increase, so that the im-provement of the performances has no additional costs in terms of energy input. For the maximum bump size, i.e. 20µm, the

42 Chapter 4. Results

pressure drop increase is 1.4% of the original one (Tab.A.2), corre-sponding to a dissipation power P = 3.72 · 10−2W, defined as the product between the mass flow rate and the pressure drop. This peculiarity is justified by the little volume variation. Let be V the T-mixer volume, Vmod the volume of the deformed geometry and

∆V = Vmod− V the volume variation. The normalized volume

variation, in case of 20 µm bump is ∆V

V =

1.32 · 10−17

2.79 · 10−10 = O 10

−8
_{, (4.1)}

which is very little because inwards and outwards displacements tend to offset themselves, so that the volume variation is negligible. Till now we have considered the optimization goal only in terms of mixing efficiency at equal volume, however the optimization

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 500 1000 1500 2000 2500 3000 δm x [µm] base geometry 4 µm 8 µm 12 µm 16 µm 20 µm

Figure 4.11: Degree of mixing trend along the mixing channel for orginal, 4, 8, 12, 16 and 20 bump size geometries

4.2. Optimized T-mixer 43 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 100 200 300 400 500 δm x [µm] base geometry 4 µm 8 µm 12 µm 16 µm 20 µm

Figure 4.12: Degree of mixing trend at the entrance of the mixing channel for orginal, 4, 8, 12, 16 and 20 bump size geometries

cess can be read with the aim of the volume reduction. Looking at Fig.4.11 and Fig.4.12, which compare the mixing degree along the outlet channel for both the original and modified geometries, it is possible to estimate the length reduction of the outlet chan-nel which ensures the same mixing degree of the original geometry. For the 20µm bump modified geometry, the mixing channel length needed to guarantee the original mixing degree is Lout= 63µm and

corresponds to a 98% decrease of the original outlet channel length. This approach, besides reducing the encumbrance, significantly re-duces the pressure drop with a decrease of 32% and a consequent reduction of the power input. All the values regarding the length and pressure reduction are reported in Tab.A.2.

44 Chapter 4. Results

4.3

Random Bump

Figure 4.13: View of the back wall for the random deformation for a 4µm bump size

0.015 0.02 0.025 0.03 0.035 0.04 0 500 1000 1500 2000 2500 3000 δm x [µm] base geometry random geometry

Figure 4.14: Degree of mixing trend along the mixing channel for the orginal and random geometries

In order to focus on the robustness of the entire procedure, we compared the performance of the optimized geometry, obtained through the application of the adjoint method, w.r.t. a geometry modified according to a sensitivity map obtained by the randomized arrangement of the sensitivity values of the previous simulation (Fig.4.13). This is a mandatory choice to not introduce effects derived by a different volume variation. The trend of the degree

4.3. Random Bump 45

of mixing along the outlet channel for the original and random geometries, for a 4µm bump size, is presented in Fig.4.14. It can be noticed how, the quality of mixing of the random geometry least differs to that of the original one with the curves almost overlap. This demonstrates the robustness of the adjoint approach and how no enhance could be obtained by a casual deformation.

Chapter 5

Conclusions

The present work, extending a previous thesis [56], has described a shape optimization procedure to improve the performance of a T-mixer. The specific objective of this study was the assessment of the potential of a shape optimization procedure based on the adjoint approach using the open-source software OpenFOAM.

An in-depth characterization of the effects of discretization on the solution has been preliminarly carried out. Although numer-ical diffusion is often said in the literature to prevent a perfectly resolution-independent solution for the T-mixer flow, to our knowl-edge dedicated studies are not available. The data collected in this work fill this gap, and we have found that, for the parameters spec-ified, a cell size of 3.0 µm is required to adequately resolve the flow field.

The shape optimization procedure, based on a sequential solu-tion of the direct flow problem, the passive scalar equasolu-tions, the adjoint solver and the passive scalar equation again, has been suc-cessfully carried out to determine the shape sensitivity. It provided excellent results, with the degree of mixing of the baseline

geom-48 Chapter 5. Conclusions

etry improved up to 222% with a single iteration of the geometry modification. Alternatively, a T-mixer with the efficiency of the baseline could be built in a more miniaturized version, by shrinking its length by as much as 98%. Mixing enhancement at the same length has been obtained without changing the volume of the flow domain, and by keeping the pressure drop essentially unchanged: in this way no penalty related to increased power is brought about by the optimized geometry. If the channel length is reduced, however, mixing enhancement and power reduction can be obtained at the same time.

The shape optimization algorithm is flexible, and can be ap-plied as is to different micromixer geometries. Particular emphasis is given to the open-source and applied nature of the software em-ployed in this work. The results of the primal flow are in perfect agreement with the ones presented in the literature and obtained through the use of commercial software. Adjoint-based optimiza-tion is becoming a viable opoptimiza-tion to be employed in applied CFD problems where standard finite-volumes codes are typically used.

Although the obtained results are promising, the effects of a number of parameters still needs to be considered in further studies: for example the optimal inlet velocity, the deformation of different wall sections of the mixer, and the study of different cost funcions. Most importantly, in view of the practical use of the new design, production technologies need to be factored in.

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Appendix A

Tables

Table A.1: Mesh Resolution Results

Cells Number Cell Length [µm] δm ∆p [kPa] δopt ∆popt[kPa]

4M 4.00 0.0334 1.46 0.133 1.47 7M 3.42 0.0324 1.46 0.123 1.47 9M 3.08 0.0421 1.46 0.116 1.47 12M 2.85 0.0406 1.46 0.112 1.47 15M 2.65 0.0397 1.46 0.108 1.47 20M 2.41 0.0385 1.46 0.102 1.47 30M 2.10 0.0369 1.46 0.096 1.47 50M 1.77 0.0355 1.46 0.088 1.47

Table A.2: Bump Size Results

bump [µm] δm ∆p [kPa] Lred[µm] ∆pred[kPa]

4 0.0775 1.464 115 0.984 8 0.0938 1.468 86 0.987 12 0.1079 1.472 66 0.989 16 0.1205 1.476 66 0.991 20 0.1278 1.481 63 0.995